Sketching parabolas and intersections

Sketching parabolas by completing the square into turning point form, and finding where a line meets a parabola by solving simultaneously.

Turning point form

Example 1●●

Sketch $y = x^2 - 6x + 5$.

Complete the square: $y = (x - 3)^2 - 4$. Turning point $(3, -4)$.

$x$-intercepts: $(x - 1)(x - 5) = 0$, so $1$ and $5$. $y$-intercept $5$.

Line and parabola

Set the equations equal. The discriminant of the resulting quadratic counts the meeting points.

Example 2●●

Find where $y = x^2$ meets $y = x + 2$.

$x^2 = x + 2$, so $x^2 - x - 2 = 0$, giving $(x - 2)(x + 1) = 0$.

Points $(2, 4)$ and $(-1, 1)$.

Flashcards

$y = (x - h)^2 + k$

Turning point $(h, k)$

Line meets parabola

Set equal, solve the quadratic.

Tangent line

$\Delta = 0$

Practice

Turning point of $y = x^2 - 6x + 5$?
$x$-intercepts of $y = x^2 - 6x + 5$?
$x$-values where $y = x^2$ meets $y = x + 2$?
Turning point of $y = x^2 + 2x - 8$?
Does $y = x - 3$ meet $y = x^2$?
For what $c$ is $y = x + c$ a tangent to $y = x^2$?
Turning point of $y = -(x - 3)^2 + 4$?
$y = x^2$ meets $y = 3x$ at the origin and at one other point. Find the other point.